The ratio of the present ages of P and Q is 7:9. Ten years ago the ratio of their ages was 5:7. What is the present age of Q in years?

Introduction / Context:
This problem is based on the concept of age ratios at two different times. The present ages of P and Q are in one ratio, and their ages ten years ago were in another ratio. Using both ratios together, we can determine their current ages. Here, we are specifically asked to find the present age of Q. Such questions strengthen understanding of how ratios and linear equations are combined in age problems.

Given Data / Assumptions:
- Present ratio of ages P:Q is 7:9.
- Ten years ago, the ratio of their ages was 5:7.
- Ages increase uniformly by 10 years over the ten year period.
- We need to find the present age of Q in years.

Concept / Approach:
We represent the present ages of P and Q using a common multiplier based on their present ratio. Then we express their ages ten years ago in terms of this multiplier and set up another ratio equation. Solving this system gives the value of the multiplier and therefore their current ages. This method relies heavily on the idea that both people age by the same fixed amount over the same period of time.

Step-by-Step Solution:
Step 1: Let the present age of P be 7k years and the present age of Q be 9k years.Step 2: Ten years ago, P was 7k - 10 years old and Q was 9k - 10 years old.Step 3: The ratio ten years ago is given as (7k - 10) : (9k - 10) = 5 : 7.Step 4: Write the equation from the ratio: (7k - 10) / (9k - 10) = 5 / 7.Step 5: Cross multiply to obtain 7 * (7k - 10) = 5 * (9k - 10).Step 6: Expand both sides: 49k - 70 = 45k - 50.Step 7: Rearrange to get 49k - 70 - 45k + 50 = 0, which simplifies to 4k - 20 = 0, so 4k = 20 and k = 5.Step 8: The present age of Q is 9k = 9 * 5 = 45 years.

Verification / Alternative check:
With k equal to 5, P is 7 * 5 = 35 years old and Q is 45 years old now. Ten years ago, P was 25 years old and Q was 35 years old. The ratio 25:35 simplifies to 5:7 when both numbers are divided by 5. The present ratio of 35:45 simplifies to 7:9 when divided by 5. Both ratios match the given conditions, confirming that Q is 45 years old.

Why Other Options Are Wrong:
If Q were 35, 25, 55, or 63 years old at present, the corresponding age of P calculated from the ratio 7:9 would not satisfy the second ratio condition from ten years ago. For example, if Q were 35 years old, P would be 35 * 7 / 9, which is not an integer age, and the ten years ago ratio would not match 5:7. Similar contradictions arise for the other options.

Common Pitfalls:
A frequent error is to attempt to work with ratios by guessing ages instead of using an algebraic approach. Another mistake is to forget to subtract 10 from both ages when forming the earlier ratio. Careful definition of variables and consistent application of the time shift are essential to avoid confusion and miscalculations.

Final Answer:
The present age of Q is 45 years.